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Abducing Relations in Continuous Spaces

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Taisuke Sato, Katsumi Inoue and Chiaki Sakama

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Proceedings of the 27th International Joint Conference on Artificial Intelligence (IJCAI-18), pages 1956-1962, 2018.
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## Abstract

We propose a new approach to abduction, i.e., non-deductive inference
to find a hypothesis H for an observation O such that H, KB |- O where
KB is background knowledge. We reformulate it linear algebraically in
vector spaces to abduce *relations*, not logical formulas, to
realize approximate but scalable abduction that can deal with
web-scale knowledge bases. More specifically we consider the problem
of abducing relations for Datalog programs with binary predicates. We
treat two cases, the non-recursive case and the recursive case.
In the non-recursive case, given r1(X,Y) and r3(X,Z), we abduce r2(Y,Z)
so that r3(X,Z) <= r1(X,Y)&r2(Y,Z) approximately holds, by computing a
matrix R2 that approximately satisfies a matrix equation R3 =
min1(R1R2) containing a nonlinear function min1(x). Here R1, R2 and
R3 encode as adjacency matrix r1(X,Y), r2(Y,Z) and r3(Y,Z)
respectively. We apply this matrix-based abduction to rule discovery
and relation discovery in a knowledge graph. The recursive case is
mathematically more involved and computationally more difficult but
solvable by deriving a recursive matrix equation and solving it. We
illustrate concrete recursive cases including a transitive closure
relation.

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